I’m working on a mathematics question and need an explanation and answer to help me learn.

After completing Units VII, I think I managed to learn how to solve linear equations involving two variables. This concept refers to a system between two equations that need to be satisfied simultaneously. Their general form can be anything such:

a1X+a2Y = m

b1X+b2Y = n

For the system to be linear, the power of the variables must not be greater than one. If we proceed to think at the solutions (X, Y) this system might have, we need to make sure that the mathematical propositions remain true when we substitute these values into the initial equations. There are a few methods we can use to find our solutions: substitution and elimination.

If we decide to use *substitution*, we should solve one of the equations for one of the variables. After that, we plug in the expression attained into the other equation. Now we have a mathematical proposition involving just one variable, which is less complicated to solve.

When it comes to *elimination*, we focus on subtracting or adding the two equations – this depends on whether the coefficients are opposed or equal –, hoping to attain the same thing: an equation with one variable.

A real-life example that I can think of – and that gave me quite some troubles when I was a child –are the problems with applicability, that contain not just numbers, but situations. In this case, we are the ones that need to create the system: figuring out the variables, coefficients, etc. But let`s take a specific example.

Suppose we want to bake 80 sweets for a festival. For this activity, we have available 130$. For a cookie, the average price is 1$ and for a muffin is 2$. Our goal is to find the number of cookies and muffins we need to bake to satisfy both requirements.

Solving:

Let C be the number of cookies and M the number of muffins. According to the problem, we can form the system of equations:

C+M = 80 (number of cookies) [1]

C+2M = 130 (dollars) [2]

Using elimination, we can subtract the first relation from the second one, obtaining:

C+2M-C-M=130-80 ó M=50

This means we need to bake 50 muffins. Going back to the first equation we have:

C+50=80 ó C=30 (cookies)

The solution for our problem is that we need to bake 50 muffins and 30 cookies.

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